THE DEVELOPMENT OF PARALLEL RESOLUTION ALGORITHMS USING THE GRAPH REPRESENTATION. Andrey Averin, Vadim Vagin
|
|
- Ashley Warren
- 5 years ago
- Views:
Transcription
1 International Journal "Information Theories & Applications" Vol THE DEVELOPMENT OF PARALLEL RESOLUTION ALGORITHMS USING THE GRAPH REPRESENTATION Andrey Averin, Vadim Vagin Abstract. The parallel resolution procedures based on graph structures method are presented. OR-, AND- and DCDP- parallel inference on connection graph representation is explored and modifications to these algorithms using heuristic estimation are proposed. The principles for designing these heuristic functions are thoroughly discussed. The colored clause graphs resolution principle is presented. The comparison of efficiency (on the Steamroller problem is carried out and the results are presented. The parallel unification algorithm used in the parallel inference procedure is briefly outlined in the final part of the paper. Keywords: Automated Reasoning, Logical inference ACM Classification Keywords: I.2.3 Artificial Intelligence: Deduction and Theorem Proving 1. Introduction The deductive inference procedures are broadly used in variety of fields, such as expert systems, decision support systems, deductive databases and intelligent information systems. Due to high amount of data in practical problems and the exponential growth of the search space, the efficiency of deductive procedures becomes the key factor in the development of deductive inference systems. One of the ways to improve the efficiency is the parallelization of inference procedures. We present a parallel inference procedure based on resolution principle. The connection graph representation is chosen as the basis for designing the parallel resolution procedures. Using the graph representation simplifies the parallelization of the inference process and allows to apply the different parallelization techniques such as OR, AND and DCDP parallelism. We study and implement OR-, AND- and DCDP- parallel resolution procedures, develop useful heuristics which can be used in parallel resolution procedures on connection graphs and make a comparison of the obtained results with the results of algorithms developed by other researchers. Also we describe and implement the clause graphs inference procedure. As the test task, the Schubert s Steamroller problem is examined [1]. The problem of parallelism on the term level is also investigated. The data structure for the term representation and the parallel unification algorithm using this data structure are presented. 2. Connection Graph The connection graph method was designed by R. Kowalski [2]. A connection graph is a scheme for representing the proper first-order formulas in disjunctive normal form. Each literal is associated with a node in the connection graph. Literals in a clause are combined into a group. If the literals in two clauses form a contrary pair (P and P then there is an edge between the respective nodes of the connection graph. Example 1. The initial set of clauses: 1.Q(c 8. F( S(y,z B(z 2. Q(b 9.B(x C(x D( 3.R(x Q( P(x 10.D(c 4.P(b 11.F(b 5. R(x S(x, T(x 12.F(c 6.T( B( 13.C(b 7.B(a
2 264 International Journal "Information Theories & Applications" Vol.13 The corresponding connection graph is shown in fig. 1. Q(c Q(b 1' 2' R ( x Q( P( x P(b 4' R ( x S( x, T ( x T ( B( 7' 3' 5' 6' B(a F ( S( y, z B( z 8' B ( x C( x D( 9' D(c 10' F(b 11' F(c C(b 12' 3. Methods of Inference on Connection Graphs Fig The Sequential Proof Procedure To prove the unsatisfiability of a clause set we must generate and resolve the initial connection graph, i.e. derive an empty clause. The main operation in connection graph refutation is the link resolution, when the resolvent is computed and added to the graph. The corresponding link is deleted and the links of the added resolvent are inserted. A pure clause is a literal group containing a node with no links. Pure clause with all its links can be easily removed from the graph without losing the completeness of the connection graph refutation procedure. Similarly, if we have a tautology clause, it also can be removed from a graph. If a resolvent on some step is a pure clause or a tautology, there is no need to insert this clause into a graph. The refutation algorithm consists of the following steps: 1. the verification whether there is any clause in a graph or not. If there are no clauses, the algorithm terminates unsuccessfully. If there is the empty clause, then the algorithm is successfully terminated, else go to step 2; 2. if a graph does not contain any link, then the algorithm is unsuccessfully terminated, else go to step 3; 3. a link selection. The link is resolved and the resolvent is generated; 4. if an empty resolvent is obtained, then the algorithm terminates successfully, else the resolvent is inserted into the graph, its links are added, and the algorithm goes to step 2. The fundamental problem in the connection graph refutation is the choice of suitable links by some criteria at each step of an algorithmic operation. Links are usually selected by using heuristics. 3.2 Parallel Inference on the Kowalski Connection Graph The Kowalski connection graph can easily be used as the basis for designing parallel resolution algorithms [3]. Since the search space is complete, there is a possibility of using parallel computation strategies for enhancing the inference procedure efficiency. Parallel resolution algorithms differ from the sequential algorithm in step 3 at which a set of links satisfying certain criteria (not a single link as in the sequential procedure is chosen and parallel resolution of all the links in this set is carried out.
3 International Journal "Information Theories & Applications" Vol OR-parallel Resolution on a Connection Graph In case of OR-parallelism, the inference system associates some goal clause with the heads of clauses possible candidates for resolution. Literals are unified and new clauses are generated. Admissible OR-links sets for the connection graph of Fig.1 are {1', 2'} and {10', 11'} DCDP-parallel Resolution on a Connection Graph One modification of the parallel inference on the Kowalski connection graph is called DCDP parallelism (parallelism for distinct clauses [4]. The correctness of the DCDP parallel resolution is proved in [5]. Definition 1. Clauses are said to be adjacent if there exist one or several links joining the literals of one clause with the literals of another clause. Definition 2. A set of links joining pairs of distinct clauses is called a DCDP-link set if the clauses of every pair are not adjacent to any clause of other pairs. To illustrate these definitions let us study the DCDP-link set for the connection graph of Fig.1. Adjacent pairs of clauses for this set are {(1, (3}, {(2, (3}, {(3, (4}, etc... Thus, one of the DCDP-link sets is {1', 6', 9'}. Other examples of DCDP-link sets are: {2', 6', 12'}, {4', 12'}, {1', 6', 10'} AND-parallel Inference Definition 3. A clause where all its links are resolved in parallel is called a SUN-clause. Definition 4. Clauses joined with the literals of a SUN-clause are called the satellite clauses. In AND-parallelism all links of literals of the SUN clause are resolved simultaneously. All resolvents are inserted in the graph along with all inherited links of a satellite clause. A SUN clause with all its links is removed. There is proved the correctness of the AND-parallel resolution [5]. The correct unification of separable variables under AND-parallel resolution is studied in [3]. Let us consider the choice of an AND-link set for the connection graph of fig.1. Admissible AND-link sets are, for example, {5', 4', 7'} (SUN-clause clause (5 and {5',6'} (SUN-clause clause (6. Detailed description of methods and algorithms can be found in [3,6]. 4. Modification of Parallel Inference Procedures Different heuristics can be used for choosing a link in resolving upon a connection graph. In the parallel resolution we must choose a set of links satisfying certain conditions. Note that the inference procedure becomes unsuitable if links are chosen unsuccessfully. The main principles underlying the design of heuristics are: 1. the number of literals in resolved clauses must be minimal, 2. the number of links in resolved clauses must be minimal, 3. the number of links in a literal for which the clauses are resolved must be minimal, 4. the unifiers of a resolved link must have a substitution of the type {c/x}, where c is a constant or a functional term, and x is a variable. Further we describe the meaning of each principle in more detail. Principle (1 simplifies a resulting connection graph, because a clause with a small number of literals, usually has a fewer number of links. Principle (2 also simplifies a resulting connection graph, because the resolution of clauses with a small number of links yields clauses also with a small number of links. Principle (3 prefers those links, the resolution of which yields "pure" clauses that on removing, can considerably simplify a connection graph.
4 266 International Journal "Information Theories & Applications" Vol.13 The same is true for principle (4: as a result of the resolution by links containing a substitution of a variable for a constant, we obtain a clause containing constant terms. Such a clause has, at the first, a small number of links and, at the second, can be effectively used in the resolution. Principles (l-(4 are taken into consideration in the heuristic function described below The Heuristic Function HI In the heuristic function H1 the link estimation is represented as a linear combination of estimations of the objects in the link (unifiers, clauses, and predicate literals in the link Computation of the Value of the Heuristic Function Let WeightLink denote the heuristic estimation of a link. Then: WeightLink= k clause (Clause 1Heur +Clause 2Heur + Uni Heur k uni + k pred (Pred 1Heur +Pred 2Heur, where Clause 1Heur,Clause 2Heur, Uni Heur, Pred 1Heur and Pred 2Heur arе the heuristic estimations for the first clause, the second clause, the link unifier, the predicate literal in the first clause of a link, and the predicate literal in the second clause of a link, respectively, and k uni, k clause and k pred [1; 100] are coefficients. Let us describe these symbols in more detail Heuristic Estimation of a Clause The heuristic estimation must take into account the changes taking place in the characteristics of a connection graph during the inference (for example, changes in the number of links and the number of literals in a clause. Let us examine the heuristic estimation based on principles (1 and (2: Clause Heur = k 1 ClauseNumberOfLinks + k 2 ClauseNumberOfClauses, where ClauseNumberOfLinks is the number of links in a clause, ClauseNumberOfClauses is the number of predicate literals in a clause; k 1, k 2 are arbitrary coefficients, chosen a priori on the basis of graph characteristics such as an average number of literals and links for the clause. Let for example the average number of literals and links for the clause be 3.2 and 4.8, respectively. Then we can take k 1 = 4.8/3.2 = 1.5 and k 2 = 1. In this case, both principles (1 and (2 have the same weight. Characteristics may change their values during the inference. In this case, the initially chosen values of coefficients become "obsolete," and one principle gains a greater weight over the other. To avoid such a situation, the values of coefficients must be changed during the inference. Let AverageLinkCount and AverageClauseLength denote an average number of links and literals in a clause, respectively. We can take k 1 = AverageLinkCount/AverageClauseLength and k 2 =1. In this case, principles (1 and (2 both gain the same weight in the course of the whole inference process. The heuristic clause estimation takes the final form: Clause Heur =(AverageLinkCount/AverageClauseLength ClauseNumberOfLinks+ ClauseNumberOfClauses Heuristic Estimation of the Unifier The heuristic unifier estimation must be based on principle 4 (the unifier of a resolved link must have a substitution of the type c/x, where с is a constant or a functional term with constants and x is a variable and must take into account the changes in values of the graph characteristics. The heuristic estimation of the unifier Uni Нeur is computed by the formula: Uni heur =AverageLinkCount/(1+NumberOfConstantSubst +NumberOfFuncSubst, where NumberOfConstantsSubst is the number of substitutions of the type {c/x} in the unifier, с is a constant term and x is a variable. NumberOfFuncSubst is the number of substitutions of the type {f/x} m the unifier, where f is a functional term with constants and x is a variable.
5 International Journal "Information Theories & Applications" Vol Heuristic Estimation of a Predicate Literal The heuristic estimation of a predicate literal must be based on principle (3 (the number of links in a literal, for which clauses are resolved upon must be minimal. The heuristic function must also take into account the changes in the graph characteristics. The heuristic estimation Pred Heur of a predicate literal is computed by the formula: Pred Heur =PredNumberOfLinks AverageClauseLength, where PredNumberOf Links is the number of links in a predicate literal Selection of Coefficients The coefficients k uni, k clause and k pred are chosen experimentally. We have chosen the following ratio k uni =k pred =(1/10 k clause for coefficients. In this relationship, the greater weight is attached to principles (1 and (2. As a rule, k clause is taken from the interval [10; 100] so that k uni and k pred lie in the interval [1; 10]. Thus, the heuristic function H1 takes account of all principles (l-(4. The weight of principles can be changed. The changes in the graph characteristics are also taken into account. The function H1 enhances the efficiency of parallel inference algorithms for problems of practical complexity. 5. Deductive Inference on Clause Graphs The deduction algorithm transforms a clause graph via two special operators a predicate node elimination operator and a predicate node splitting operator [6,7]. They are applied to a predicate vertex depending on whether the node has multiarcs or not. A predicate node I said to be joined to a clause with multiarcs if the clause contains more than one literal with predicate symbol of the predicate node. For example, the node P in Fig.2 is joined to clause 1,2 and 3 (clause 4 is joined with the predicate node P with an ordinary arc by the multiarcs. 1. P( x, P(f(x, ( or P(x, P(f(x, 2. Pu (, f( x & Pvgw (, ( ( or P(u,f(x P(v,g(w 3. P(g(x, P(a,z (or P(g(x, P(a,z 4. P(a,x ( or P(a,x 2 P(v,g(w P(u,f(x P P(x, 1 P(f(x, P(a,x P(g(x, P(a,z 4 3 Fig. 2 In this method clauses are expressed in implicative form. Here 1-4 are the nodes corresponding to the clauses of the logical model. The oval vertex represents the predicate symbol P. Continuous arcs are weighted with conditional literals of clauses, whereas dotted ars are weighted with inference literals of the clause. An operation similar to colouring of clause graphs is introduced. Clause condition is «network is coloured» by color C1 (the corresponding arcs in figures are shown by a continuous line and the condition for inference is colored by color C2 (a dotted line in figures. If clause graphs are represented in colored form, then it is easy to
6 268 International Journal "Information Theories & Applications" Vol.13 search for information and new assertions can be inferred easily and thus the effectiveness of the inference system is enhanced. The general inference scheme for an empty clause via transformation of clause graphs consists of the following: 1. if the network contains predicate nodes to which the node elimination operator can be applied, then such nodes are removed by this elimination operator; 2. if there are nodes with multiarcs, then the splitting operator is applied to generate nodes without multiarcs. Then the node elimination operator is applied, etc. until a contradiction is obtained in network. If the node elimination operator consists of a set of usual operations of resolution of specially chosen pairs of clauses, then the splitting operator contains a distinct feature of this algorithm, i.e., a feature that has no analog in other logical inference mechanisms. The predicate node elimination operator is applied to nodes having no multiarcs. It resolves in all possible ways those clauses that contain contrary pairs of literals. Resolution is implemented by the predicate of the chosen predicate node. Upon completion of all resolutions, parent clauses and the predicate node are removed from the network and the new clauses found via resolution are included in the network. Figures 3a and 3b show a clause graph before and after the application of the elimination operator to the vertex M for the following disjunct set: 1. Q & M 2. M & H Q 3. F & M H 4. M 5. F The node M and clauses (1-(4 were removed and the clauses 6. Q. (1,4 7. F. H 8H Q (3,4 (2,4 were added to the network. Q 1 M M M F F H Q F F H Q M F H Q F H Q M H Q H 2 8 а b Fig. 3 The predicate splitting operator is introduced in order to remove multiarcs from nodes and thereby create conditions for the application of the elimination operator. The splitting operator generates copies of clauses and a few new vertices having the same predicate symbol as the split vertex, but with new indexes 1,2,3,. The clause copies and new nodes are interrelated with each other. First, the elimination operator removes those nodes that have higher indexes. Figure 4 shows the action of a splitting operator for the case of three multiarcs starting from clauses 1,2 and 3 (Fig. 2.
7 International Journal "Information Theories & Applications" Vol ' 3' 4'' 4''' P 1 ( a, x P1 ( g( x, P ( a, 2 z P ( a, 2 x P ( a, 3 z P ( a, P ( g( x, 3 x 3''' 2 P( v, g( w 4 P( u, f ( x 3 y P( a, x P 1 P2 P 3 P P( g( x, P1 ( u, f ( x P( f ( x, P2 ( u, f ( x P3 ( v, g( w P 3( v, g( w P3 ( x, P( a, z 2' 2'' 1''' 3 P1 ( x, P2 ( x, 1'' P( f ( x, P( f ( x, 1' Fig.4 Splitting the three multiarcs ( petals of the node P by this operator 1we obtain four nodes P 1, P 2, P 3 and P (their amount is one more than the amount of «petals», four copies of clause 4, three copies of the «split» clause 1, two copies of the «split» clause 2 with a duplicate at vertex P, and a «split» clause 3 with a duplicate at nodes P and P 3. Splitting occurs in the following order: first the multiarc (1 P, then the multiarc (2 P, and finally the multiarc (3 P are split. The priority elimination strategies for removing nodes having no multiarcs and nodes with minimal number of arcs are reasonable for forming a sequence of processed nodes in a clause graph. They yield a sequence of eliminations and generate a lesser number of intermediate clauses than the variant in which these strategies are not used. Both these operations are correct i.e. unsatisfiable clauses remain unsatisfiable after the application of these operators[1]. A parallel algorithm for deductive inference on colored clause graphs is described in [7,8]. 5. Efficiency As the test problem, we have researched the "steamroller" problem formulated by L. Schubert in 1978 to test automatic proof systems [1]. This problem requires the generation of an exponential number of intermediate clauses and unifications during the inference process. Below we describe the results obtained by our and other algorithms of deductive inference for the «steamroller» problem. procedures have also shown their efficiency for deductive inference on connection graphs. CG is the inference strategy with a connection graph. SOS is the inference on a connection graph with a goal statement as a support set. TR is the inference within Theory Links - the extension of the standard resolution method. UR is inference with Unit Resolution - a modification of the resolution method. LUR is the inference by Linked Unit Resolution - a modification of the UR resolution method. According to Figs. 5 and 6, the best results are obtained by the McCune/LUR procedures. Parallel inference
8 270 International Journal "Information Theories & Applications" Vol Successful unification attempts Stickel/CG-SOS-TR Stickel/CG Stickel/CG-TR McCune/UR McCune/LUR Colored semantic networks inference DCDP-parallel inference on connection graphs AND-parallel inference on connection graphs OR-parallel inference on connection graphs Sequential inference on connection graphs Fig Intermediate clauses Stickel/CG-SOS-TR Stickel/CG Stickel/CG-TR McCune/UR McCune/LUR Colored semantic networks inference DCDP-parallel inference on connection graphs AND-parallel inference on connection graphs OR-parallel inference on connection graphs Sequential inference on connection graphs Fig Parallel Unification in Connection Graph Inference Due to a high amount of the unification tasks in deductive inference, the efficiency of the implementation of the unification procedure is one of the main factors in designing deductive inference procedures. There is a variety of effective unification procedures, but all of them have their own disadvantages. The main disadvantage is the use of complex, tightly coupled data structures, which are very difficult to parallelize. We develop rather simple data structure for the term representation based on the notion of path strings. Terms (and clauses are stored in tables, which are connected with links. Some tables and parts of the tables can be processed in parallel (with having links in mind. As we have no possibility to describe the algorithm and the data structure for term representation in detail, we just briefly outline the main principles used in this approach. The term is stored as the sequence of strings, where every string presents one symbol in the term. Also such information as the type of a symbol (a variable or a constant symbol, the index number of an argument in a function symbol and the depth of a symbol (i.e. the number of function symbols which this symbol belongs to is stored. The terms that can be unified are connected with the links of two types. The first type corresponds to the terms (and strings of different depth. The second type corresponds to the terms of the same depth. Let us illustrate these notions by the simple example. Consider the unification task {t 1 =t 2 }, where t 1 = f(a,g(h(x and t 2 =f(x,g(. The representation of the term f(a,g(h(x is f1.a.1.constant(1 and f2.g1.h1.x.3.variable (2.
9 International Journal "Information Theories & Applications" Vol The representation of the term f(x,g( is f1.x.1.variable(3 and f2.g1.x.2.variable(4 (the index number is shown in brackets. The links are created between the strings (1 and (3 and between the strings (2 and (4. The first link has the second type and the second link has the first type. The task of link establishing is one of the main tasks in the proposed approach. Two techniques can be used. The simplest one is the special sorting procedure on tables containing strings from the unification task. The main drawback of this approach is the deficit of efficiency caused by the nature of the sorting problem (the complexity of the sorting problem is nlog(n. The second approach is based on automata representation of strings and is similar to discrimination trees. Using automata increases the efficiency of the procedure, but requires higher memory consumption. The main idea lying in the parallelization of the unification procedure is the use of dependencies graph, where strings connected with arcs cannot be proceeded in parallel. The complexity of the task of determining maximum sets for parallel proceeding is equal to the complexity of the graph coloring task, though heuristics can be used to find non-maximum but satisfactory sets. This unification procedure has been combined with the parallel inference procedure on connection graphs. The structure for term representation and unification are thoroughly described in [9]. 7. Conclusions The Kowalski connection graph and the sequential inference procedure on connection graphs are investigated. The structure of the OR-, AND- and DCDP-parallel inference methods is described. Methods of modifying parallel inference procedures are analyzed and the main principles underlying the design of heuristic link estimations are stated. The heuristic function for link selection is designed. A procedure of inference on colored clause graphs is also described. The developed deductive inference procedures are compared with other procedures on the "steamroller" problem. The procedures of parallel inference on connection graphs and clause graphs are the effective ones for the deductive inference. They can be applied in expert and decision making systems. Bibliography [1] M.F. Stickel, Schubert's Steamroller Problem: Formulations and Solutions // J. Autom. Reasoning, 1986, vol. 2, pp [2] R. Kowalski, A Proof Procedure using Connection Graphs // J. ACM, 1975, 22(4, pp [3] V.N. Vagin and N.O. Salapina, A System of Parallel Inference with the Use of a Connection Graph // Journal of Computer and System Sciences International, Vol. 37, No. 5, 1998, pp [4] G. Hornung, A. Knapp and U. Knapp, A Parallel Connection Graph Proof Procedure // German Workshop on Artificial Intelligence. Lecture Notes in Computer Science, Berlin: Springer-Verlag, pp [5] R. Loganantharaj, Theoretical and Implemetational Aspects of Parallel Link Resolution in Connection Graphs, Ph.D. Thesis, Dept. of Computer Science, Colorado State Univ., [6] Vagin V.N. Deduktsiya i obobshcenie v sistemakh prinyatia reshenii (Deduction and Generalization in Decision Making Systems, Moscow: Nauka, [7] Vagin V.N., Parallel Inference on Logical Networks, IFIP/WG 12.3 International Workshop on Automated Reasoning, Beijing, China, 1992, pp [8] A.I. Averin, V.N. Vagin and M.K. Khamidulov, The Investigation of Algorithms of Parallel Inference by the Steamroller Problem // Journal of Computer and System Sciences International, Vol. 39, No. 5, 2000, pp [9] A.I. Averin, V.N. Vagin, Using Parallelism in Deductive Inference, // Journal of Computer and System Sciences International vol. 45, 4. pp [10] Vagin V.N., Golovina E.Y., Zagoryanzkaya A.A., Fomina M.V.. «Dostoverniy i pravdopodobnyi vivod v intellektualnykh systemakh». (Relaible and Plausible Inference in Intellectual Systems Моscow: Fizmatlit Authors' Information Andrey Averin Moscow Power Engineering Institute, Krasnokasarmennaya str., 14, Moscow, Russia; averin@rbcmail.ru Vadim Vagin Moscow Power Engineering Institute, Krasnokasarmennaya str., 14, Moscow, Russia; vagin@apmsun.mpei.ac.ru
AND-OR GRAPHS APPLIED TO RUE RESOLUTION
AND-OR GRAPHS APPLIED TO RUE RESOLUTION Vincent J. Digricoli Dept. of Computer Science Fordham University Bronx, New York 104-58 James J, Lu, V. S. Subrahmanian Dept. of Computer Science Syracuse University-
More informationNotes for Chapter 12 Logic Programming. The AI War Basic Concepts of Logic Programming Prolog Review questions
Notes for Chapter 12 Logic Programming The AI War Basic Concepts of Logic Programming Prolog Review questions The AI War How machines should learn: inductive or deductive? Deductive: Expert => rules =>
More informationPropositional Resolution Part 3. Short Review Professor Anita Wasilewska CSE 352 Artificial Intelligence
Propositional Resolution Part 3 Short Review Professor Anita Wasilewska CSE 352 Artificial Intelligence Resolution Strategies We present here some Deletion Strategies and discuss their Completeness. Deletion
More informationA MECHANIZATION OF TYPE THEORY. Gerard P. HUBT IRIA - LABORIA Rocquencourt FRANCE
Session 6 Logic: II Theorem Proving and A MECHANIZATION OF TYPE THEORY Gerard P. HUBT IRIA - LABORIA Rocquencourt FRANCE ABSTRACT A refutational system of logic for a language of order w ia presented.
More informationCONSIDERATIONS CONCERNING PARALLEL AND DISTRIBUTED ARCHITECTURE FOR INTELLIGENT SYSTEMS
CONSIDERATIONS CONCERNING PARALLEL AND DISTRIBUTED ARCHITECTURE FOR INTELLIGENT SYSTEMS 1 Delia Ungureanu, 2 Dominic Mircea Kristaly, 3 Adrian Virgil Craciun 1, 2 Automatics Department, Transilvania University
More informationModule 6. Knowledge Representation and Logic (First Order Logic) Version 2 CSE IIT, Kharagpur
Module 6 Knowledge Representation and Logic (First Order Logic) 6.1 Instructional Objective Students should understand the advantages of first order logic as a knowledge representation language Students
More informationMathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras. Lecture - 37 Resolution Rules
Mathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras Lecture - 37 Resolution Rules If some literals can be unified, the same algorithm should be able
More informationFoundations of AI. 9. Predicate Logic. Syntax and Semantics, Normal Forms, Herbrand Expansion, Resolution
Foundations of AI 9. Predicate Logic Syntax and Semantics, Normal Forms, Herbrand Expansion, Resolution Wolfram Burgard, Andreas Karwath, Bernhard Nebel, and Martin Riedmiller 09/1 Contents Motivation
More informationIntegrity Constraints (Chapter 7.3) Overview. Bottom-Up. Top-Down. Integrity Constraint. Disjunctive & Negative Knowledge. Proof by Refutation
CSE560 Class 10: 1 c P. Heeman, 2010 Integrity Constraints Overview Disjunctive & Negative Knowledge Resolution Rule Bottom-Up Proof by Refutation Top-Down CSE560 Class 10: 2 c P. Heeman, 2010 Integrity
More informationCSL105: Discrete Mathematical Structures. Ragesh Jaiswal, CSE, IIT Delhi
is another way of showing that an argument is correct. Definitions: Literal: A variable or a negation of a variable is called a literal. Sum and Product: A disjunction of literals is called a sum and a
More informationA proof-producing CSP solver: A proof supplement
A proof-producing CSP solver: A proof supplement Report IE/IS-2010-02 Michael Veksler Ofer Strichman mveksler@tx.technion.ac.il ofers@ie.technion.ac.il Technion Institute of Technology April 12, 2010 Abstract
More informationAn Improvement on Sub-Herbrand Universe Computation
12 The Open Artificial Intelligence Journal 2007 1 12-18 An Improvement on Sub-Herbrand Universe Computation Lifeng He *1 Yuyan Chao 2 Kenji Suzuki 3 Zhenghao Shi 3 and Hidenori Itoh 4 1 Graduate School
More informationROUGH MEMBERSHIP FUNCTIONS: A TOOL FOR REASONING WITH UNCERTAINTY
ALGEBRAIC METHODS IN LOGIC AND IN COMPUTER SCIENCE BANACH CENTER PUBLICATIONS, VOLUME 28 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1993 ROUGH MEMBERSHIP FUNCTIONS: A TOOL FOR REASONING
More informationComputational Logic. SLD resolution. Damiano Zanardini
Computational Logic SLD resolution Damiano Zanardini UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid damiano@fi.upm.es Academic Year 2009/2010
More informationBoolean Representations and Combinatorial Equivalence
Chapter 2 Boolean Representations and Combinatorial Equivalence This chapter introduces different representations of Boolean functions. It then discusses the applications of these representations for proving
More informationPropositional Logic. Part I
Part I Propositional Logic 1 Classical Logic and the Material Conditional 1.1 Introduction 1.1.1 The first purpose of this chapter is to review classical propositional logic, including semantic tableaux.
More informationGraph Theory Questions from Past Papers
Graph Theory Questions from Past Papers Bilkent University, Laurence Barker, 19 October 2017 Do not forget to justify your answers in terms which could be understood by people who know the background theory
More informationLinear Clause Generation by Tableaux and DAGs
kovasznai 2007/8/10 11:27 page 109 #1 5/1 (2007), 109 118 tmcs@inf.unideb.hu http://tmcs.math.klte.hu Linear Clause Generation by Tableaux and DAGs Gergely Kovásznai Abstract. Clause generation is a preliminary
More informationThe Resolution Algorithm
The Resolution Algorithm Introduction In this lecture we introduce the Resolution algorithm for solving instances of the NP-complete CNF- SAT decision problem. Although the algorithm does not run in polynomial
More informationTidying up the Mess around the Subsumption Theorem in Inductive Logic Programming Shan-Hwei Nienhuys-Cheng Ronald de Wolf bidewolf
Tidying up the Mess around the Subsumption Theorem in Inductive Logic Programming Shan-Hwei Nienhuys-Cheng cheng@cs.few.eur.nl Ronald de Wolf bidewolf@cs.few.eur.nl Department of Computer Science, H4-19
More informationInfinite Derivations as Failures
Infinite Derivations as Failures Andrea Corradi and Federico Frassetto DIBRIS, Università di Genova, Italy name.surname@dibris.unige.it Abstract. When operating on cyclic data, programmers have to take
More informationLogic: TD as search, Datalog (variables)
Logic: TD as search, Datalog (variables) Computer Science cpsc322, Lecture 23 (Textbook Chpt 5.2 & some basic concepts from Chpt 12) June, 8, 2017 CPSC 322, Lecture 23 Slide 1 Lecture Overview Recap Top
More informationSystem Description: iprover An Instantiation-Based Theorem Prover for First-Order Logic
System Description: iprover An Instantiation-Based Theorem Prover for First-Order Logic Konstantin Korovin The University of Manchester School of Computer Science korovin@cs.man.ac.uk Abstract. iprover
More informationAutomatic Reasoning (Section 8.3)
Automatic Reasoning (Section 8.3) Automatic Reasoning Can reasoning be automated? Yes, for some logics, including first-order logic. We could try to automate natural deduction, but there are many proof
More information[Ch 6] Set Theory. 1. Basic Concepts and Definitions. 400 lecture note #4. 1) Basics
400 lecture note #4 [Ch 6] Set Theory 1. Basic Concepts and Definitions 1) Basics Element: ; A is a set consisting of elements x which is in a/another set S such that P(x) is true. Empty set: notated {
More informationPROPOSITIONAL LOGIC (2)
PROPOSITIONAL LOGIC (2) based on Huth & Ruan Logic in Computer Science: Modelling and Reasoning about Systems Cambridge University Press, 2004 Russell & Norvig Artificial Intelligence: A Modern Approach
More informationOn 2-Subcolourings of Chordal Graphs
On 2-Subcolourings of Chordal Graphs Juraj Stacho School of Computing Science, Simon Fraser University 8888 University Drive, Burnaby, B.C., Canada V5A 1S6 jstacho@cs.sfu.ca Abstract. A 2-subcolouring
More informationOnline algorithms for clustering problems
University of Szeged Department of Computer Algorithms and Artificial Intelligence Online algorithms for clustering problems Summary of the Ph.D. thesis by Gabriella Divéki Supervisor Dr. Csanád Imreh
More informationTerm Algebras with Length Function and Bounded Quantifier Elimination
with Length Function and Bounded Ting Zhang, Henny B Sipma, Zohar Manna Stanford University tingz,sipma,zm@csstanfordedu STeP Group, September 3, 2004 TPHOLs 2004 - p 1/37 Motivation: Program Verification
More information2SAT Andreas Klappenecker
2SAT Andreas Klappenecker The Problem Can we make the following boolean formula true? ( x y) ( y z) (z y)! Terminology A boolean variable is a variable that can be assigned the values true (T) or false
More informationThe Resolution Principle
Summary Introduction [Chang-Lee Ch. 5.1] for Propositional Logic [Chang-Lee Ch. 5.2] Herbrand's orem and refutation procedures Satisability procedures We can build refutation procedures building on Herbrand's
More informationLogic Programming and Resolution Lecture notes for INF3170/4171
Logic Programming and Resolution Lecture notes for INF3170/4171 Leif Harald Karlsen Autumn 2015 1 Introduction This note will explain the connection between logic and computer programming using Horn Clauses
More informationStorage and Retrieval of First Order Logic Terms in a Database
Storage and Retrieval of First Order Logic Terms in a Database Peter Gurský Department of Computer Science, Faculty of Science P.J.Šafárik University Košice Jesenná 9, 040 01, Košice gursky@vk.science.upjs.sk
More informationA Partial Correctness Proof for Programs with Decided Specifications
Applied Mathematics & Information Sciences 1(2)(2007), 195-202 An International Journal c 2007 Dixie W Publishing Corporation, U. S. A. A Partial Correctness Proof for Programs with Decided Specifications
More informationCLAW-FREE 3-CONNECTED P 11 -FREE GRAPHS ARE HAMILTONIAN
CLAW-FREE 3-CONNECTED P 11 -FREE GRAPHS ARE HAMILTONIAN TOMASZ LUCZAK AND FLORIAN PFENDER Abstract. We show that every 3-connected claw-free graph which contains no induced copy of P 11 is hamiltonian.
More informationConstruction of a transitive orientation using B-stable subgraphs
Computer Science Journal of Moldova, vol.23, no.1(67), 2015 Construction of a transitive orientation using B-stable subgraphs Nicolae Grigoriu Abstract A special method for construction of transitive orientations
More informationXI International PhD Workshop OWD 2009, October Fuzzy Sets as Metasets
XI International PhD Workshop OWD 2009, 17 20 October 2009 Fuzzy Sets as Metasets Bartłomiej Starosta, Polsko-Japońska WyŜsza Szkoła Technik Komputerowych (24.01.2008, prof. Witold Kosiński, Polsko-Japońska
More informationAssignment 4 Solutions of graph problems
Assignment 4 Solutions of graph problems 1. Let us assume that G is not a cycle. Consider the maximal path in the graph. Let the end points of the path be denoted as v 1, v k respectively. If either of
More informationKnowledge Representation and Reasoning Logics for Artificial Intelligence
Knowledge Representation and Reasoning Logics for Artificial Intelligence Stuart C. Shapiro Department of Computer Science and Engineering and Center for Cognitive Science University at Buffalo, The State
More informationsketchy and presupposes knowledge of semantic trees. This makes that proof harder to understand than the proof we will give here, which only needs the
The Subsumption Theorem in Inductive Logic Programming: Facts and Fallacies Shan-Hwei Nienhuys-Cheng Ronald de Wolf cheng@cs.few.eur.nl bidewolf@cs.few.eur.nl Department of Computer Science, H4-19 Erasmus
More informationArtificial Intelligence
Artificial Intelligence Graph theory G. Guérard Department of Nouvelles Energies Ecole Supérieur d Ingénieurs Léonard de Vinci Lecture 1 GG A.I. 1/37 Outline 1 Graph theory Undirected and directed graphs
More informationPost and Jablonsky Algebras of Compositions (Superpositions)
Pure and Applied Mathematics Journal 2018 (6): 95-100 http://www.sciencepublishinggroup.com/j/pamj doi: 10.11648/j.pamj.2018006.13 SSN: 2326-990 (Print) SSN: 2326-9812 (Online) Post and Jablonsky Algebras
More informationModule 6. Knowledge Representation and Logic (First Order Logic) Version 2 CSE IIT, Kharagpur
Module 6 Knowledge Representation and Logic (First Order Logic) Lesson 15 Inference in FOL - I 6.2.8 Resolution We have introduced the inference rule Modus Ponens. Now we introduce another inference rule
More informationFinding Rough Set Reducts with SAT
Finding Rough Set Reducts with SAT Richard Jensen 1, Qiang Shen 1 and Andrew Tuson 2 {rkj,qqs}@aber.ac.uk 1 Department of Computer Science, The University of Wales, Aberystwyth 2 Department of Computing,
More informationOn Algebraic Expressions of Generalized Fibonacci Graphs
On Algebraic Expressions of Generalized Fibonacci Graphs MARK KORENBLIT and VADIM E LEVIT Department of Computer Science Holon Academic Institute of Technology 5 Golomb Str, PO Box 305, Holon 580 ISRAEL
More informationDecision Procedures for Recursive Data Structures with Integer Constraints
Decision Procedures for Recursive Data Structures with Ting Zhang, Henny B Sipma, Zohar Manna Stanford University tingz,sipma,zm@csstanfordedu STeP Group, June 29, 2004 IJCAR 2004 - p 1/31 Outline Outline
More informationAnswers to specimen paper questions. Most of the answers below go into rather more detail than is really needed. Please let me know of any mistakes.
Answers to specimen paper questions Most of the answers below go into rather more detail than is really needed. Please let me know of any mistakes. Question 1. (a) The degree of a vertex x is the number
More informationOrthogonal art galleries with holes: a coloring proof of Aggarwal s Theorem
Orthogonal art galleries with holes: a coloring proof of Aggarwal s Theorem Pawe l Żyliński Institute of Mathematics University of Gdańsk, 8095 Gdańsk, Poland pz@math.univ.gda.pl Submitted: Sep 9, 005;
More informationBOOLEAN ALGEBRA AND CIRCUITS
UNIT 3 Structure BOOLEAN ALGEBRA AND CIRCUITS Boolean Algebra and 3. Introduction 3. Objectives 3.2 Boolean Algebras 3.3 Logic 3.4 Boolean Functions 3.5 Summary 3.6 Solutions/ Answers 3. INTRODUCTION This
More informationWeighted Geodetic Convex Sets in A Graph
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. PP 12-17 www.iosrjournals.org Weighted Geodetic Convex Sets in A Graph Jill K. Mathew 1 Department of Mathematics Mar Ivanios
More informationPROBLEM SET 1 SOLUTIONS MAS341: GRAPH THEORY 1. QUESTION 1
PROBLEM SET 1 SOLUTIONS MAS341: GRAPH THEORY 1. QUESTION 1 Find a Hamiltonian cycle in the following graph: Proof. Can be done by trial an error. Here we find the path using some helpful observations.
More informationOperational Semantics
15-819K: Logic Programming Lecture 4 Operational Semantics Frank Pfenning September 7, 2006 In this lecture we begin in the quest to formally capture the operational semantics in order to prove properties
More informationI-CONTINUITY IN TOPOLOGICAL SPACES. Martin Sleziak
I-CONTINUITY IN TOPOLOGICAL SPACES Martin Sleziak Abstract. In this paper we generalize the notion of I-continuity, which was defined in [1] for real functions, to maps on topological spaces. We study
More informationSome bounds on chromatic number of NI graphs
International Journal of Mathematics and Soft Computing Vol.2, No.2. (2012), 79 83. ISSN 2249 3328 Some bounds on chromatic number of NI graphs Selvam Avadayappan Department of Mathematics, V.H.N.S.N.College,
More informationMetric Dimension in Fuzzy Graphs. A Novel Approach
Applied Mathematical Sciences, Vol. 6, 2012, no. 106, 5273-5283 Metric Dimension in Fuzzy Graphs A Novel Approach B. Praba 1, P. Venugopal 1 and * N. Padmapriya 1 1 Department of Mathematics SSN College
More informationMulti Domain Logic and its Applications to SAT
Multi Domain Logic and its Applications to SAT Tudor Jebelean RISC Linz, Austria Tudor.Jebelean@risc.uni-linz.ac.at Gábor Kusper Eszterházy Károly College gkusper@aries.ektf.hu Abstract We describe a new
More informationTHE FOUNDATIONS OF MATHEMATICS
THE FOUNDATIONS OF MATHEMATICS By: Sterling McKay APRIL 21, 2014 LONE STAR - MONTGOMERY Mentor: William R. Brown, MBA Mckay 1 In mathematics, truth is arguably the most essential of its components. Suppose
More informationThis is a preprint of an article accepted for publication in Discrete Mathematics c 2004 (copyright owner as specified in the journal).
This is a preprint of an article accepted for publication in Discrete Mathematics c 2004 (copyright owner as specified in the journal). 1 Nonorientable biembeddings of Steiner triple systems M. J. Grannell
More informationLogic and its Applications
Logic and its Applications Edmund Burke and Eric Foxley PRENTICE HALL London New York Toronto Sydney Tokyo Singapore Madrid Mexico City Munich Contents Preface xiii Propositional logic 1 1.1 Informal introduction
More informationConnecting face hitting sets in planar graphs
Connecting face hitting sets in planar graphs Pascal Schweitzer and Patrick Schweitzer Max-Planck-Institute for Computer Science Campus E1 4, D-66123 Saarbrücken, Germany pascal@mpi-inf.mpg.de University
More informationTwo Characterizations of Hypercubes
Two Characterizations of Hypercubes Juhani Nieminen, Matti Peltola and Pasi Ruotsalainen Department of Mathematics, University of Oulu University of Oulu, Faculty of Technology, Mathematics Division, P.O.
More informationAn LCF-Style Interface between HOL and First-Order Logic
An LCF-Style Interface between HOL and First-Order Logic Joe Hurd Computer Laboratory University of Cambridge, joe.hurd@cl.cam.ac.uk 1 Introduction Performing interactive proof in the HOL theorem prover
More informationOnline Appendix: A Stackelberg Game Model for Botnet Data Exfiltration
Online Appendix: A Stackelberg Game Model for Botnet Data Exfiltration June 29, 2017 We first provide Lemma 1 showing that the urban security problem presented in (Jain et al. 2011) (see the Related Work
More informationA New Reduction from 3-SAT to Graph K- Colorability for Frequency Assignment Problem
A New Reduction from 3-SAT to Graph K- Colorability for Frequency Assignment Problem Prakash C. Sharma Indian Institute of Technology Survey No. 113/2-B, Opposite to Veterinary College, A.B.Road, Village
More informationCOMP4418 Knowledge Representation and Reasoning
COMP4418 Knowledge Representation and Reasoning Week 3 Practical Reasoning David Rajaratnam Click to edit Present s Name Practical Reasoning - My Interests Cognitive Robotics. Connect high level cognition
More informationSymmetric Product Graphs
Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 5-20-2015 Symmetric Product Graphs Evan Witz Follow this and additional works at: http://scholarworks.rit.edu/theses
More informationPart I Logic programming paradigm
Part I Logic programming paradigm 1 Logic programming and pure Prolog 1.1 Introduction 3 1.2 Syntax 4 1.3 The meaning of a program 7 1.4 Computing with equations 9 1.5 Prolog: the first steps 15 1.6 Two
More informationNew Worst-Case Upper Bound for #2-SAT and #3-SAT with the Number of Clauses as the Parameter
Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence (AAAI-10) New Worst-Case Upper Bound for #2-SAT and #3-SAT with the Number of Clauses as the Parameter Junping Zhou 1,2, Minghao
More informationNP-Completeness of 3SAT, 1-IN-3SAT and MAX 2SAT
NP-Completeness of 3SAT, 1-IN-3SAT and MAX 2SAT 3SAT The 3SAT problem is the following. INSTANCE : Given a boolean expression E in conjunctive normal form (CNF) that is the conjunction of clauses, each
More informationLeveraging Transitive Relations for Crowdsourced Joins*
Leveraging Transitive Relations for Crowdsourced Joins* Jiannan Wang #, Guoliang Li #, Tim Kraska, Michael J. Franklin, Jianhua Feng # # Department of Computer Science, Tsinghua University, Brown University,
More informationEfficient Two-Phase Data Reasoning for Description Logics
Efficient Two-Phase Data Reasoning for Description Logics Abstract Description Logics are used more and more frequently for knowledge representation, creating an increasing demand for efficient automated
More informationExact Algorithms Lecture 7: FPT Hardness and the ETH
Exact Algorithms Lecture 7: FPT Hardness and the ETH February 12, 2016 Lecturer: Michael Lampis 1 Reminder: FPT algorithms Definition 1. A parameterized problem is a function from (χ, k) {0, 1} N to {0,
More informationDeductive Methods, Bounded Model Checking
Deductive Methods, Bounded Model Checking http://d3s.mff.cuni.cz Pavel Parízek CHARLES UNIVERSITY IN PRAGUE faculty of mathematics and physics Deductive methods Pavel Parízek Deductive Methods, Bounded
More informationFundamental Properties of Graphs
Chapter three In many real-life situations we need to know how robust a graph that represents a certain network is, how edges or vertices can be removed without completely destroying the overall connectivity,
More informationInsensitive Traffic Splitting in Data Networks
Juha Leino and Jorma Virtamo. 2005. Insensitive traffic splitting in data networs. In: Proceedings of the 9th International Teletraffic Congress (ITC9). Beijing, China, 29 August 2 September 2005, pages
More informationEfficient Circuit to CNF Conversion
Efficient Circuit to CNF Conversion Panagiotis Manolios and Daron Vroon College of Computing, Georgia Institute of Technology, Atlanta, GA, 30332, USA http://www.cc.gatech.edu/home/{manolios,vroon} Abstract.
More informationOn Resolution Proofs for Combinational Equivalence Checking
On Resolution Proofs for Combinational Equivalence Checking Satrajit Chatterjee Alan Mishchenko Robert Brayton Department of EECS U. C. Berkeley {satrajit, alanmi, brayton}@eecs.berkeley.edu Andreas Kuehlmann
More informationThe crossing number of K 1,4,n
Discrete Mathematics 308 (2008) 1634 1638 www.elsevier.com/locate/disc The crossing number of K 1,4,n Yuanqiu Huang, Tinglei Zhao Department of Mathematics, Normal University of Hunan, Changsha 410081,
More informationResolution (14A) Young W. Lim 6/14/14
Copyright (c) 2013-2014. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free
More informationthe application rule M : x:a: B N : A M N : (x:a: B) N and the reduction rule (x: A: B) N! Bfx := Ng. Their algorithm is not fully satisfactory in the
The Semi-Full Closure of Pure Type Systems? Gilles Barthe Institutionen for Datavetenskap, Chalmers Tekniska Hogskola, Goteborg, Sweden Departamento de Informatica, Universidade do Minho, Braga, Portugal
More informationA Pearl on SAT Solving in Prolog (extended abstract)
A Pearl on SAT Solving in Prolog (extended abstract) Jacob M. Howe and Andy King 1 Introduction The Boolean satisfiability problem, SAT, is of continuing interest because a variety of problems are naturally
More informationFormally-Proven Kosaraju s algorithm
Formally-Proven Kosaraju s algorithm Laurent Théry Laurent.Thery@sophia.inria.fr Abstract This notes explains how the Kosaraju s algorithm that computes the strong-connected components of a directed graph
More informationLecture 17 of 41. Clausal (Conjunctive Normal) Form and Resolution Techniques
Lecture 17 of 41 Clausal (Conjunctive Normal) Form and Resolution Techniques Wednesday, 29 September 2004 William H. Hsu, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Reading: Chapter 9,
More informationOne-Point Geometric Crossover
One-Point Geometric Crossover Alberto Moraglio School of Computing and Center for Reasoning, University of Kent, Canterbury, UK A.Moraglio@kent.ac.uk Abstract. Uniform crossover for binary strings has
More informationThe Encoding Complexity of Network Coding
The Encoding Complexity of Network Coding Michael Langberg Alexander Sprintson Jehoshua Bruck California Institute of Technology Email: mikel,spalex,bruck @caltech.edu Abstract In the multicast network
More informationQUTE: A PROLOG/LISP TYPE LANGUAGE FOR LOGIC PROGRAMMING
QUTE: A PROLOG/LISP TYPE LANGUAGE FOR LOGIC PROGRAMMING Masahiko Sato Takafumi Sakurai Department of Information Science, Faculty of Science University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, JAPAN
More informationDesigning Views to Answer Queries under Set, Bag,and BagSet Semantics
Designing Views to Answer Queries under Set, Bag,and BagSet Semantics Rada Chirkova Department of Computer Science, North Carolina State University Raleigh, NC 27695-7535 chirkova@csc.ncsu.edu Foto Afrati
More informationCharacterization of Request Sequences for List Accessing Problem and New Theoretical Results for MTF Algorithm
Characterization of Request Sequences for List Accessing Problem and New Theoretical Results for MTF Algorithm Rakesh Mohanty Dept of Comp Sc & Engg Indian Institute of Technology Madras, Chennai, India
More informationSolutions for the Exam 6 January 2014
Mastermath and LNMB Course: Discrete Optimization Solutions for the Exam 6 January 2014 Utrecht University, Educatorium, 13:30 16:30 The examination lasts 3 hours. Grading will be done before January 20,
More informationOn Computing the Minimal Labels in Time. Point Algebra Networks. IRST { Istituto per la Ricerca Scientica e Tecnologica. I Povo, Trento Italy
To appear in Computational Intelligence Journal On Computing the Minimal Labels in Time Point Algebra Networks Alfonso Gerevini 1;2 and Lenhart Schubert 2 1 IRST { Istituto per la Ricerca Scientica e Tecnologica
More informationKnowledge Representation and Reasoning Logics for Artificial Intelligence
Knowledge Representation and Reasoning Logics for Artificial Intelligence Stuart C. Shapiro Department of Computer Science and Engineering and Center for Cognitive Science University at Buffalo, The State
More informationA Simplified Correctness Proof for a Well-Known Algorithm Computing Strongly Connected Components
A Simplified Correctness Proof for a Well-Known Algorithm Computing Strongly Connected Components Ingo Wegener FB Informatik, LS2, Univ. Dortmund, 44221 Dortmund, Germany wegener@ls2.cs.uni-dortmund.de
More informationFast and Simple Algorithms for Weighted Perfect Matching
Fast and Simple Algorithms for Weighted Perfect Matching Mirjam Wattenhofer, Roger Wattenhofer {mirjam.wattenhofer,wattenhofer}@inf.ethz.ch, Department of Computer Science, ETH Zurich, Switzerland Abstract
More informationThe Inverse of a Schema Mapping
The Inverse of a Schema Mapping Jorge Pérez Department of Computer Science, Universidad de Chile Blanco Encalada 2120, Santiago, Chile jperez@dcc.uchile.cl Abstract The inversion of schema mappings has
More informationPACKING DIGRAPHS WITH DIRECTED CLOSED TRAILS
PACKING DIGRAPHS WITH DIRECTED CLOSED TRAILS PAUL BALISTER Abstract It has been shown [Balister, 2001] that if n is odd and m 1,, m t are integers with m i 3 and t i=1 m i = E(K n) then K n can be decomposed
More informationAnswer Key #1 Phil 414 JL Shaheen Fall 2010
Answer Key #1 Phil 414 JL Shaheen Fall 2010 1. 1.42(a) B is equivalent to B, and so also to C, where C is a DNF formula equivalent to B. (By Prop 1.5, there is such a C.) Negated DNF meets de Morgan s
More information(a) (4 pts) Prove that if a and b are rational, then ab is rational. Since a and b are rational they can be written as the ratio of integers a 1
CS 70 Discrete Mathematics for CS Fall 2000 Wagner MT1 Sol Solutions to Midterm 1 1. (16 pts.) Theorems and proofs (a) (4 pts) Prove that if a and b are rational, then ab is rational. Since a and b are
More informationQuantification. Using the suggested notation, symbolize the statements expressed by the following sentences.
Quantification In this and subsequent chapters, we will develop a more formal system of dealing with categorical statements, one that will be much more flexible than traditional logic, allow a deeper analysis
More informationRecognizing Interval Bigraphs by Forbidden Patterns
Recognizing Interval Bigraphs by Forbidden Patterns Arash Rafiey Simon Fraser University, Vancouver, Canada, and Indiana State University, IN, USA arashr@sfu.ca, arash.rafiey@indstate.edu Abstract Let
More informationCSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Sections p.
CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Sections 10.1-10.3 p. 1/106 CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer
More information